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Selected Differential System Examples from Lectures. w i. V = Ah. w o. Liquid Storage Tank. Standing assumptions Constant liquid density r Constant cross-sectional area A Other possible assumptions Steady-state operation Outlet flow rate w 0 known function of liquid level h. - PowerPoint PPT Presentation
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Selected Differential System Examples from Lectures
Liquid Storage Tank
Standing assumptions» Constant liquid density » Constant cross-sectional area A
Other possible assumptions» Steady-state operation» Outlet flow rate w0 known function of liquid level h
V = Ah
wi
wo
Mass Balance
Mass balance on tank
Steady-state operation:
Valve characteristics
Linear ODE model
Nonlinear ODE model
0
out
0
inonaccumulati
)(ww
dt
dhAww
dt
Ahdii
ii wwww 000
hCwhCw vovo NonlinearLinear
0)0( hhhCwdt
dhA vi
0)0( hhhCwdt
dhA vi
Stirred Tank Chemical Reactor
Assumptions» Pure reactant A in feed stream
» Perfect mixing
» Constant liquid volume
» Constant physical properties (, k)
» Isothermal operation
A
k
kCr
BA
Overall mass balance
Component balance
qqqqdt
Vdii
0)(
0)0()(
)(
AAAAAiA
AAAiiA
CCkCCCV
q
dt
dC
VkCqCCqdt
VCd
Plug-Flow Chemical Reactor
Assumptions» Pure reactant A in feed stream» Perfect plug flow» Steady-state operation» Isothermal operation» Constant physical properties (, k)
A
k
kCr
BA
z
qi, CAi qo, CAoCA(z) z
Plug-Flow Chemical Reactor cont.
Overall mass balance
qqqdz
dqz
i
zzz
z
zzz
0
0
out Massin Mass
0
0)()(
lim
0)()(
z
qi, CAi qo, CAoCA(z) z
Component balance
AiAAA
AA
AzzAzA
z
AzzAzA
CCkCdz
dC
A
q
kCdz
dC
A
q
kCz
CC
A
q
zAkCqCqC
)0(0
0
0)()(
lim
0)()(
0
consumedA outA inA
Continuous Biochemical Reactor
Fresh Media Feed (substrates)
Exit Gas Flow
Agitator
Exit Liquid Flow(cells & products)
Cell Growth Modeling
Specific growth rate
Yield coefficients» Biomass/substrate: YX/S = -X/S» Product/substrate: YP/S = -P/S» Product/biomass: YP/X = P/X» Assumed to be constant
Substrate limited growth
» S = concentration of rate limiting substrate» Ks = saturation constant» m = maximum specific growth rate (achieved when S >> Ks)
(g/L)ion concentrat biomass 1
Xdt
dX
X
SK
SS
S
m
)(
Continuous Bioreactor Model
Assumptions Sterile feed Constant volume Perfect mixing Constant temperature and pH Single rate limiting nutrient Constant yields Negligible cell death
Product formation rates» Empirically related to specific growth rate
» Growth associated products: q = YP/X» Nongrowth associated products: q = » Mixed growth associated products: q = YP/X
Mass Balance Equations
Cell mass
» VR = reactor volume» F = volumetric flow rate» D = F/VR = dilution rate
Product
Substrate
» S0 = feed concentration of rate limiting substrate
XDXdt
dXXVFX
dt
dXV RR
qXDPdt
dPqXVFP
dt
dPV RR
XY
SSDdt
dSXV
YFSFS
dt
dSV
SXR
SXR
/0
/0
1)(
1
Exothermic CSTR
Scalar representation
Vector representation
00 )0()exp()( AAAAAfA CCCRTEkCC
V
q
dt
dC
00 )0(
)()/exp()()( TT
VC
TTUA
C
CRTEkHTT
V
q
dt
dT
p
c
p
Af
0
0
0
0
)0()(
)()/exp()(
)(
)/exp()()(
T
C
dt
d
TTVC
UA
C
CRTEkHTT
V
q
CRTEkCCV
q
T
C
A
cpp
Af
AAAfA
yyfy
yfy
)(
/exp)( 0
TTUAQ
RTEkTk
BA
c
k
Isothermal Batch Reactor
CSTR model: A B C
Eigenvalue analysis: k1 = 1, k2 = 2
Linear ODE solution:
0)0(,10)0(211 BABAB
AA CCCkCk
dt
dCCk
dt
dC
1
11
1
02
21
01
)(
)()(
)2(2
)1(1 xx
A
yydt
dy
tC
tCty
B
A
tttt
B
A ecececectC
tCt
1
1
1
0
)(
)()( 2
21
)2(2
)1(1
21 xxy
Isothermal Batch Reactor cont.
Linear ODE solution:
Apply initial conditions:
Formulate matrix problem:
Solution:
tt
B
A ecectC
tCt
1
1
1
0
)(
)()( 2
21y
0
10
1
1
1
0
)0(
)0()0( 21 cc
C
C
B
Ay
10
10
0
10
11
10
2
1
2
1
c
c
c
c
tt
ttt
B
A
ee
eee
tC
tC
1010
10
1
110
1
010
)(
)(2
2
Isothermal CSTR
Nonlinear ODE model
Find steady-state point (q = 2, V = 2, Caf = 2, k = 0.5)
)(2)(2 2AAAAf
Ak CfkCCCV
q
dt
dCBA
12
31
)1)(2(
)2)(1)(4(11
02
0)5.0)(2()2(2
22)()(
2
2
22
A
AA
AAAAAfA
C
CC
CCCkCCV
qCf
Isothermal CSTR cont.
Linearize about steady-state point:
This linear ODE is an approximation to the original nonlinear ODE
'''
''
3])5.0)(2)(2([
0
)(
AAAAA
A
CAA
A
CCCCdt
dC
CC
fCf
dt
dC
A
Continuous Bioreactor
Cell mass balance
Product mass balance
Substrate mass balance
XDXdt
dX
qXDPdt
dP
XY
SSDdt
dS
SX
/
0
1)(
Steady-State Solutions
Simplified model equations
Steady-state equations
Two steady-state points
),()(1
)(
)(),()(
2/
0
1
SXfXSY
SSDdt
dSSK
SSSXfXSDX
dt
dX
SX
S
m
0)(1
)(
)(0)(
/0
XSY
SSD
SK
SSXSXD
SX
S
m
0:Washout
)()(:Trivial-Non
0
0/
XSS
SSYXD
DKSDS SX
m
S
Model Linearization
Biomass concentration equation
Substrate concentration equation
Linear model structure:
SSK
SX
SK
XXDS
SSS
fXX
X
fSXf
dt
Xd
S
m
S
m
SXSX
2
,
1
,
11
zero
),(
SDSK
SX
SK
X
YX
SK
S
Y
SSS
fXX
X
fSXf
dt
Sd
S
m
S
m
SXS
m
SX
SXSX
2//
,
2
,
22
11
zero
),(
SaXadt
Sd
SaXadt
Xd
2221
1211
Non-Trivial Steady State
Parameter values» KS = 1.2 g/L, m
= 0.48 h-1, YX/S = 0.4 g/g
» D = 0.15 h-1, S0 = 20 g/L
Steady-state concentrations
Linear model coefficients (units h-1)
529.31
375.01
472.10
2/
22/
21
21211
DSK
SX
SK
X
Ya
SK
S
Ya
SK
SX
SK
Xaa
S
m
S
m
SXS
m
SX
S
m
S
m
g/L 78.7)(g/L 545.0 0/
SSYXD
DKS SX
m
S
Stability Analysis
Matrix representation
Eigenvalues (units h-1)
Conclusion» Non-trivial steady state is asymptotically stable» Result holds locally near the steady state
Axxdt
dx
S
Xx
529.3375.0
472.10
365.3164.0529.3375.0
472.111
IA
Washout Steady State
Steady state: Linear model coefficients (units h-1)
Eigenvalues (units h)
Conclusion» Washout steady state is unstable» Suggests that non-trivial steady state is globally stable
15.01
132.11
0303.0
2
maxmax
/22
max
/21
12
max
11
DSK
SX
SK
X
Ya
SK
S
Ya
aDSK
Sa
SSSXSSX
S
g/L 0g/L 20 XSS i
15.0303.015.0132.1
0303.011
IA
Gaussian Quadrature Example
Analytical solution
Variable transformation
Approximate solution
Approximation error = 4x10-3%
067545.2125
1
5
1
21
21
xx edxe
32)(2
23
21
txxab
baxt
066691.21)533346.10(2
533346.10)55555.0()88889.0()55555.0(
22
5
1
1
1
77459.0077459.0
1
1
1
1
)32(5
1
21
23
23
23
23
23
21
21
dxe
eeedte
dtedtedxe
x
t
ttx
Plug-Flow Reactor Example
Ai
N
NAA
AiAnAnA
nAnAnA
AiAAA
CqzkA
zCLC
CzCzCqzkA
zC
zkCz
zCzC
A
q
CCkCdz
dC
A
q
1
1)()(
)()(1
1)(
0)()()(
)0(0
01
11
A
k
kCr
BA
z
qi, CAi qo, CAoCA(z) z
0 L
Plug-Flow Reactor Example cont.
Analytical solution
Numerical solution
Convergence formula
Convergence of numerical solution
Ai
N
Ai
N
A CNqkAL
CqzkA
LC
1
1
1
1)(
L
q
kACLC AiA exp)(
Lq
kACC
NqLkA AiAi
N
Nexp
1
1lim
a
N
Ne
Na
1
1lim
Matlab Example
Isothermal CSTR model
Model parameters: q = 2, V = 2, Caf = 2, k = 0.5
Initial condition: CA(0) = 2
Backward Euler formula
Algorithm parameters: h = 0.01, N = 200
)(2)(2 2AAAAf
Ak CfkCCCV
q
dt
dCBA
)(2)( ,,2
,,,1, nAnAnAnAAfnAnA ChfCkCCCV
qhCC
Matlab Implementation: iso_cstr_euler.m
h = 0.01;
N = 200;
Cao = 2;
q = 2;
V = 2;
Caf = 2;
k = 0.5;
t(1) = 0;
Ca(1) = Cao;
for i=1:N
t(i+1) = t(i)+h;
f = q/V*(Caf-Ca(i))-2*k*Ca(i)^2;
Ca(i+1)= Ca(i)+h*f;
end
plot(t,Ca)
ylabel('Ca (g/L)')
xlabel('Time (min)')
axis([0,2,0.75,2.25])
Euler Solution
>> iso_cstr_euler
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.8
1
1.2
1.4
1.6
1.8
2
2.2
CA
(g/
L)
Time (min)
Solution with Matlab Function
function f = iso_cstr(x)
Cao = 2;
q = 2;
V = 2;
Caf = 2;
k = 0.5;
Ca = x(1);
f(1) = q/V*(Caf-Ca)-2*k*Ca^2;
>> xss = fsolve(@iso_cstr,2)
xss = 1.0000
>> df = @(t,x) iso_cstr(x);
>> [t,x] = ode23(df,[0,2],2);
>> plot(t,x)
>> ylabel('Ca (g/L)')
>> xlabel('Time (min)')
>> axis([0,2,0.75,2.25])
Matlab Function Solution
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Ca
(g/L
)
Time (min)
Euler
Matlab
CSTR Example
Van de Vusse reaction
CSTR model
Forward Euler
233
2211
3
21
2 Ar
BArr
CkrDA
CkrCkrCBA
0221
012
31
)0(),(
)0(),(2)(
BBBABABB
AABAAAAAiA
CCCCfCkCkCV
q
dt
dC
CCCCfCkCkCCV
q
dt
dC
00,,2,1,,,,2,1,
00,2
,3,1,,,,1,1,
),(
2)(),(
BBnBnAnBnBnBnAnBnB
AAnAnAnAAinAnBnAnAnA
CCCkCkCV
qhCCChfCC
CCCkCkCCV
qhCCChfCC
Stiff System Example
CSTR model: A B C
Homogeneous system:
Eigenvalue analysis: q/V = 1, k1 = 1, k2 = 200
BABB
AAAiA CkCkC
V
q
dt
dCCkCC
V
q
dt
dC211)(
BBBAAA CtCtCCtCtC )()()()( ''
'2
'1
''
'1
''
BABB
AAA CkCkC
V
q
dt
dCCkC
V
q
dt
dC
2012
''2011
02'
)(
)()('
21
'
'
yydt
dy
tC
tCty
B
A A
Explicit Solution
Forward Euler
First iterative equation
Second iterative equation
)201(201
)2(2
',
',
',
'1,
'''
',
',
'1,
''
nBnAnBnBBAB
nAnAnAAA
CChCCCCdt
dC
ChCCCdt
dC
unstable andy oscillator1
stablebut y oscillator1
behaved well0
)21()21()2(
21
21
'0,
',
',
',
',
'1,
h
h
h
ChCChChCC An
nAnAnAnAnA
unstable andy oscillator
stablebut y oscillator
behaved well0
)2011()21()201(
2012
2012
2011
2011
'0,
'0,
',
',
',
',
'1,
h
h
h
ChhChCCChCC Bn
An
nBnBnAnBnB
Implicit Solution
Backward Euler
First iterative equation
Second iterative equation
)201(201
)2(2
'1,
'1,
',
'1,
'''
'1,
',
'1,
''
nBnAnBnBBAB
nAnAnAAA
CChCCCCdt
dC
ChCCCdt
dC
behaved well0)21(
1)2( '
0,'
,'
1,'
,'
1,
h
Ch
CChCC AnnAnAnAnA
behaved well0)2011(
1
)21(
1)201( '
0,'
0,1'
,'
1,'
1,'
,'
1,
h
Ch
hCh
CCChCC BnAnnBnBnAnBnB
Matlab Solution
function f = stiff_cstr(x)
Cai = 2;
qV = 1;
k1 = 1;
k2 = 200;
Ca = x(1);
Cb = x(2);
f(1) = qV*(Cai-Ca)-k1*Ca;
f(2) = -qV*Cb+k1*Ca-k2*Cb;
f = f';
>> xo = fsolve(@stiff_cstr,[1 1])
xo = 1.0000 0.0050
>> df = @(t,x) stiff_cstr(x);
>> [t,x] = ode23(df,[0,2],[2 0]);
>> [ts,xs] = ode23s(df,[0,2],[2 0]);
>> size(t)
ans = 173 1
>> size(ts)
ans = 30 1
Matlab Solution cont.
>> subplot(2,1,1)
>> plot(t,x(:,1))
>> hold
Current plot held
>> plot(ts,xs(:,1),'r')
>> ylabel('Ca (g/L)')
>> ylabel('Ca (g/L)')
>> xlabel('Time (min)')
>> legend('ode23','ode23s')
>> subplot(2,1,2)
>> plot(t,x(:,2))
>> hold
Current plot held
>> plot(ts,xs(:,2),'r')
>> ylabel('Cb (g/L)')
>> xlabel('Time (min)')
>> legend('ode23','ode23s')
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21
1.5
2
Ca
(g/L
)
Time (min)
ode23
ode23s
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.005
0.01C
b (g
/L)
Time (min)
ode23
ode23s
Binary Flash Unit
Schematic diagram
Vapor-liquid equilibrium
x
xy
)1(1
FeedF, xF
Liquid
V, y
L, x
Vapor
QHeatH
Assumptions» Saturated liquid feed» Perfect mixing» Negligible heat losses» Negligible vapor holdup» Negligible energy
accumulation» Constant heat of
vaporization» Constant relative
volatility
Model Formulation
Mass balance
Component balance
)()()(
)()(
)(
xyVxxFLxVyFxxLVFdt
dxH
dt
dxHLVFx
dt
dxH
dt
dHx
dt
Hxd
LxVyFxdt
Hxd
FF
F
LVFdt
dH
Model Formulation cont.
Steady-state energy balance
Index 0 DAE model
vv H
QVQHV
x
xy
xyH
HQxx
H
F
dt
dxF
)1(10
)(/
)(
),(0
),(
yxg
yxfdt
dx
Binary Flash Unit Revisited
DAE model
Parameter values: H = 5, F = 10, xF = 0.5, V = 2, = 10
x
xy
xyH
Vxx
H
F
dt
dxF
)1(10
)()(
Solver Problems Method
ode15s Stiff DAEs up to index 1 Numerical differentiation
ode23t Moderately stiff DAEs up to index 1
Trapezoidal
MATLAB DAE Solution Codes
binary_flash.m
function f = binary_flash(x)
H = 5;
F = 10;
xf = 0.5;
V = 2;
alpha = 10;
xv = x(1);
yv = x(2);
f(1) = F/H*(xf-xv)-V/H*(yv-xv);
f(2) = yv-alpha*xv/(1+(alpha-1)*xv);
f = f';
Matlab Commands
Results for V = 2
>> f = @(x) binary_flash(x);
>> xss = fsolve(f,[1 1],[])
xss = 0.4068 0.8727
>> df = @(t,x) binary_flash(x);
>> M = [1 0; 0 0];
>> options=odeset('Mass',M);
>> [t1,y1]=ode15s(df,[0,10],xss,options);
Change V = 1
>> [t2,y2]=ode15s(df,[0,10],xss,options);
Results for V = 2 and V = 1
0 1 2 3 4 5 6 7 8 9 100.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Time
Mol
e F
ract
ion
Liquid
Vapor